Let's say we have a curvilinear coordinate system $(\rho,\theta,\zeta)$. Also, let's say we have a smooth and closed surface $\Gamma$ parameterized as $\Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\zeta) = \mathbf{S}(\theta,\zeta)$. Let $\mathbf{n}$ be the unit vector normal to the surface $\Gamma$.
Is $\nabla \times \mathbf{n} = 0 $ guaranteed for any $\Gamma$ that is closed and smooth?
I know there's a similar question about this here: Curl of unit normal vector on a surface is zero?
However, I do not know if you can write $\mathbf{n}$ as $\frac{\nabla \phi}{|\nabla \phi|}$ for any smooth and closed surface.